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Resistance of the double random phase encryption against various attacks

Yann Frauel, Albertina Castro, Thomas J. Naughton, and Bahram Javidi

Open Access Open Access

Abstract

Several attacks are proposed against the double random phase encryption scheme. These attacks are demonstrated on computer-generated ciphered images. The scheme is shown to be resistant against brute force attacks but susceptible to chosen and known plaintext attacks. In particular, we describe a technique to recover the exact keys with only two known plain images. We compare this technique to other attacks proposed in the literature.

©2007 Optical Society of America

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Figures (6)
Fig. 1.
Fig. 1. Principle of the double random phase encryption scheme.

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Fig. 2.
Fig. 2. Decryption of images encrypted with 16-level phase keys. (a)–(b) 16-level decryption key, (c)–(d) 4-level decryption key, and (e)–(f) 2-level decryption key.

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Fig. 3.
Fig. 3. Decryption using partial windows of the original 100×100 key. (a)–(b) 50×50 window, (c)–(d) 40×40 window, and (e)–(f) 30×30 window.

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Fig. 4.
Fig. 4. Decryption using partial windows of the original 100×100 key and reduction to three phase levels. (a)–(b) 50×50 window, (c)–(d) 40×40 window, and (e)–(f) 30×30 window.

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Fig. 5.
Fig. 5. Decryption using Attack 9. (a) and (b) are the two known plain images. (c) Ciphered image corresponding to an unknown plain image. (d) Unknown image decrypted using the keys retrieved by Attack 9.

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Fig. 6.
Fig. 6. Decryption of noisy ciphered images using an adaptation of Attack 9. (a) Plain images. (b) Decrypted images using the keys retrieved by simple system solving. (c) Decrypted images using the keys retrieved by least-square solving.

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Equations (26)

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C = Y · 𝓕 ( P · X ) ,
P · X = 𝓕 − 1 ( C ÷ Y )
P = ∣ P · X ∣ = ∣ 𝓕 − 1 ( C ÷ Y ) ∣ ,
P 3 · X = 𝓕 − 1 ( C 3 ÷ Y ) ,
X ∝ 𝓕 − 1 ( C 3 ÷ Y )
C = T P ,
C = T P ,
T = C P − 1 ,
P = T − 1 C ,
c i = y i ∑ j = 1 N F ij x j p j with 1 ≤ i ≤ N ,
c i ′ = y i ∑ j = 1 N F ij x j p j ′ with 1 ≤ i ≤ N .
c i ∑ j = 1 N F ij x j p j ′ = c i ′ ∑ j = 1 N F ij x j p j with 1 ≤ i ≤ N ,
∑ j = 1 N F ij x j ( c i p j ′ − c i ′ p j ) = 0 with 1 ≤ i ≤ N .
∑ j = 1 N S ij x j = 0 with 1 ≤ i ≤ N ,
∑ j = 1 N − 1 S ij x j = − S iN with 1 ≤ i ≤ N − 1 .
y i = c i ∑ j = 1 N F ij x j p j with 1 ≤ i ≤ N .
∑ j = 1 N S ij αβ x j = 0 with 1 ≤ i ≤ N ,
S αβ X = 0 ,
[ S 12 S 13 S 14 S 23 S 24 S 34 0 … 0 1 ] x = [ 0 ⋮ 0 1 ] ,
x ̂ i = x i ∣ x i ∣ with 1 ≤ i ≤ N .
λ i α y i = c i α with 1 ≤ i ≤ N ,
λ i α = ∑ j = 1 N F ij x ̂ j p j α .
Λ α Y = C α ,
Λ α = [ λ 1 α ⋱ λ N α ] .
[ Λ 1 Λ 2 Λ 3 Λ 4 ] Y = [ C 1 C 2 C 3 C 4 ] .
y ̂ i = y i ∣ y i ∣ with 1 ≤ i ≤ N .
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